Differential Equations

The Geometric View of ODEs

01Direction Fields and the Geometric View of ODEs 02Integral Curves and the Isocline Method 03Perpendicular Isoclines: The Circles of y' = -x/y 04Isocline Corridors and a Trap for Solutions 05Existence and Uniqueness: One Curve Through a Point 06Where Existence and Uniqueness Fail Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

Numerical Methods

01Euler's Method and Its Three Equations 02Euler's Method by Hand: Building the Table 03Convexity and the Sign of Euler's Error 04First Order Error: Halve the Step, Halve the Error 05Improved Euler (RK2): Averaging Two Slopes 06Runge-Kutta 4 and the Price of Accuracy 07Singular Points: Where Every Method Fails Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

First-Order Linear Equations

01First Order Linear: Standard Form and the Sign of p 02The Conduction-Diffusion Model: Newton's Law of Cooling 03The Integrating Factor and the Four-Step Method 04Integrating Factors: Two Mistakes and a Trick Question 05Constant Coefficients: Steady State and Transient Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Integrating Factors: Two Mistakes and a Trick Question

Which two habits — skipping standard form and forgetting to multiply $q$ — quietly wreck an integrating-factor solution?

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Your summary note

1. Skipping standard form and forgetting $q$
  Record both pitfalls: finding $p$ before dividing through by the coefficient of $y'$ , and multiplying only the left side by the integrating factor.
2. Standard form and the integrating factor $p$
  Divide $(1+\cos x)y' - (\sin x)y = 2x$ through by $1+\cos x$ , integrate $p = -\frac{\sin x}{1+\cos x}$ to $\ln(1+\cos x)$ , and exponentiate.
3. Multiplying back returns the original equation
  Show that the standard-form equation times $1+\cos x$ reproduces the equation you started with, then use the product rule to write its left side as $[(1+\cos x)y]'$ .
4. Finishing the solution with $y(0) = 1$
  Integrate both sides to $(1+\cos x)y = x^2 + c$ , solve for $y$ , substitute $x = 0$ to get $c = 2$ , and state $y = \frac{x^2+2}{1+\cos x}$ .
Attempt 1 of 2